2. Asymptotic Expansion for Large Conductivity and Skin Effect
Let
be a bounded region in
representing a metallic conductor and
represent air. The parameters
,
,
denoting permittivity, permeability and conductivity, are assumed to be zero in
with positive
,
and
values in
. Let the incident electric and magnetic fields,
and
, satisfy Maxwell’s equations in air. The total fields
and
satisfy the same Maxwell’s equations as
and
in
, but a different set of equations in
. Across the interface
, which is assumed to be a regular analytic surface, the tangential components of both
and
are continuous.
and
represent the scattered fields. All fields are time-harmonic with frequency
. As in [
1], we neglect conduction (displacement) currents in air (metal).
Then, with appropriate scaling, the eddy current problem is (see [
6,
7]).
Problem : Given
and
, find
and
, such that
Here and are dimensionless parameters, and , if displacement currents are neglected in metal . The subscript T denotes a tangential component, and the superscripts plus and minus denote limits from and .
At higher frequencies, the constant is usually large, leading to the perfect conductor approximation. Formally this means solving only the equation and requiring that on . If we let and denote the scattered fields, we obtain
Problem : Given
, find
and
, such that
Remark 1. There exists at most one solution of problem for any and (see [8]). Remark 2. There exists a sequence , such that if , then curl, in , on Σ implies in .
We are interesting in an asymptotic expansion of the solution of problem
with respect to inverse powers of conductivity. With
denoting the distance from
measured into
along the normal to
, the expansions reads:
Here
and
are independent of
, which is proportional to
. The exponential in (
5) and (
6) represents the skin effect. Next, we present from [
1] these expansions for the half-space case where the various coefficients can be computed recursively. Note
and
in (
3) and (
4), respectively, are simply the perfect conductor approximation, that is, the solution of
.
and
in (
3) and (
4) can be calculated successively by solving a sequence of problems of the same form as
but with boundary values determined from earlier coefficients. The
and
in (
5) and (
6), respectively, are obtained by solving ordinary differential equations in the variable
.
For the ease of the reader, we present here in the half-space case
, i.e.,
, and
, i.e.,
, a formal procedure to compute
,
, which was given by MacCamy and Stephan [
1]. They substituted Equations (
3)–(
6) into
for
and equated coefficients of
. Here, we give a short description of their approach.
Let
and decompose field
into tangential and normal components:
with orthogonal component
, and unit vectors
(
).
Then, one computes with the surface gradient
, the rotation
and
Now, by setting
, one obtains for
and
Hence, matching coefficients of and , respectively, yields , and implying .
As coefficients of
, one obtains
Now the gauge condition implies and ; hence and Thus, .
Equating coefficients of
in (
11) gives
MacCamy and Stephan obtained in [
1] with
,
,
:
and
and
For
, we have that
yields
Matching coefficients of
, one finds in
(and corresponding due to
)
With the above relations, the recursion process goes as follows. First one use (6.10) for
and (6.13), in [
1], to conclude that
Now
is just the solution of
, which can be solved by the boundary integral equation procedure introduced in MacCamy and Stephan [
1] and revisited below. However, from
we obtain
Now, the right side of (
17) is known and easily computed. Then
and (
17) yield
Therefore, by (6.10), in [
1], we have, again, a new solvable problem for
which is just like
, that is
but with new boundary values for
as given by (
18).
For the complete algorithm see [
1]. Note, with
, we have
yielding in
A comparison with Peron’s results (see Chapter 5 in [
9]) shows that
,
, in
,
and
. Furthermore, we see that the first terms in the asymptotic expansion of the electrical field for a smooth surface
derived by Peron coincide with those for the half-space
investigated by MacCamy and Stephan, namely,
,
,
.
Remark 3. From Theorem 5 in Chapter 3 of [10], there exists only one solution to the electromagnetic transmission problem for a smooth interface. This solution which can be computed by the boundary integral equation procedure is shown below, where we assume that (19) holds. Then, for the electrical field obtained via the boundary integral equation system, we have that in the tubular region , there holds for the remainders obtained by truncating (3) and (5) at for constants , independent of ρ. 3. A Boundary Integral Equation Method of the First Kind
Next, we describe the integral equation procedure for
and
from [
1,
7,
11,
12]. Throughout the section, we require that
These methods, like others, are based on the Stratton–Chu formulas from [
6]. To describe these, some notation is needed. Let
denote the exterior normal to
. Given any vector field
defined on
, we have
where
, which lies in the tangent plane, is the tangential component of
.
Define the simple layer potential
for density
(correspondingly for a vector field) for the surface
by
For a vector field
on
, define
by (
21) with
replacing
.
We collect in the following lemma some of the well-known results about the simple layer potential .
Remark 4 (Lemma 2.1 in [
1]).
For any complex κ, and any continuous ψ on Σ, there holds:- (i)
is continuous in ,
- (ii)
in ,
- (iii)
as ,
- (iv)
where as . - (v)
where the matrix function satisfies as .
For problem
in
, the Stratton–Chu formula gives
Similarly, for problem
, in
For given
,
and
, (
23) yields a solution of
. However, we know only
. The standard treatment of
starts from (
23), sets
and
and replaces
with an unknown tangential field
yielding
Then the boundary condition yields an integral equation of the second kind for in the tangent space to .
The method (
24) is analogous to solving the Dirichlet problem for the scalar Helmholtz equation with a double layer potential. However, having found
, it is hard to determine
, or equivalently
, on
. Note that calculating
on
involves finding a second normal derivative of
.
The method in [
1] for
is analogous to solving the scalar problems with a simple layer potential (see [
13]). MacCamy and Stephan use (
23), but this time they set
and replace
and
by unknowns
and
M. Thus, they take
If they can determine , then in this case, they can use Remark 4 to determine ; hence, on .
With the surface gradient
on
, the boundary conditions in (
1) and (
25) imply, by continuity of
,
or equivalently,
Note that for any field
defined in a neighborhood of
, one can define the surface divergence
by
As shown in [
1]), there holds, for any differentiable tangential field
, that
Setting
on
yields, therefore, with (
25),
and
gives immediately
5. Galerkin Procedure for the Perfect Conductor Problem ()
Next, we present implementations of the Galerkin methods (see [
7,
10,
19,
20]) and some numerical experiments for the integral equations (
26) and (
27). These experiments were performed with the package
Maiprogs (cf. Maischak [
21,
22]), which is a Fortran-based program package utilized for finite element and boundary element simulations [
23]. Initially developed by M. Maischak,
Maiprogs has been extended for electromagnetic problems by Teltscher [
24] and Leydecker [
25].
We investigate the exterior problem
by performing the integral equations procedure with (
26) and (
27):
Testing against arbitrary functions
and
in (
26) and (
27), we get
Partial integration in the second term of
shows that the formulation (
35) is symmetric: by definition of symmetric bilinear forms
a and
c, of the bilinear form
b and linear form
ℓ through
the variational formulation has the form: find
such that
for all
.
We now work with finite dimensional subspaces
of dimension
n and
of dimension
m, and seek approximations
and
for
and
M, such that
for all
and
.
Let
be a basis of
and
be a basis of
.
, and
are of the forms
Inserting (
38) in (
37) provides
for all
and
,
,
.
With matrices and vectors
(
39) has also the form
We have considered a basis of and a basis of . These functions were chosen as piecewise polynomials. To obtain these bases, we considered suitable basis functions locally on the element of a grid, i.e., on each component grid.
Start from a grid
with
N elements, and let
and
be the basis of a square reference element
. The local basis functions on an element
are each
or
.
Therefore, we should calculate first
where
or
are the basis functions of
and
Test each local basis function against any other local basis function and sum the result to the test value of the global basis functions, which include these local basis functions.
Let be the index set for the grid elements, the index set for the basic functions on the reference element and the index set for the global basis functions.
Let be the mapping from local to global basis functions, such that , if the local basis function component of the global basis function is .
Let
be the set of all pairs of
with
; then,
We are dealing in this implementation with Raviart–Thomas basis functions. The transformation of these functions requires a Peano transformation
. Thus, if
,
is calculated by
, then the Peano transformation of the local basis functions to the basic functions on the reference element then gives
with
and
, and referent element
.
The calculation of the integrals with Helmholtz kernel
is not exact. We consider the expansion of the Helmholtz kernel in a Taylor series. There holds
The first terms are singular for
, and their corresponding integrals are treated by analytic evaluation in
Maiprogs (cf. Maischak [
21,
22,
26]), but the integrals of all other terms can be calculated with sufficient accuracy by Gaussian quadrature.
Compute
with
described above, and
, the analogously defined map for the basic functions of
.
While a transformation of the scalar basis functions is not required, the transformation of the surface divergence of Raviart–Thomas elements is carried out by
and we have
with
and
. The calculation of
is similar to the one mentioned before.
The calculation of the right-hand side appears simple at first glance, since there are no single layer potential terms. However, the right-hand side must be computed by quadrature.
The quadrature of an integral over
on the reference element is determined by the quadrature points
, and the associated weights
, which are processed in
x and
y directions. Perform the two-dimensional quadrature as a combination of one-dimensional quadratures in each
x and
y direction, and use here the weights from the already implemented one-dimensional quadrature formula. With
quadrature points in
x-direction and
quadrature points in
y-direction, the quadrature formula reads:
The quadrature points on the square reference element and the corresponding weights for Gaussian quadrature were implemented in Maiprogs already. For triangular elements, use a Duffy transformation.
We will now calculate the right-hand side in the Galerkin formulation, i.e., the linear form
ℓ, applied to the base functions
,
. The quadrature takes place on the reference element. Decompose global functions into local basis functions and then use the Peano transformation for the Raviart–Thomas functions. Therefore,
with
. Applying (
45) with
, leads to
with
. As before, the task is carried out by looping through all grid components, and the values are added to the entries for each of its base function.
The electrical field can be calculated by
We have for the first term in (
47) with
Then using Peano transformation, it follows that
For the second term in (
47), one gets
The calculation of
is done as follows (compare Remark 4
).
6. Numerical Experiments
Here, consider one example to test the implementation. As the domain, take the cube
. We tested the Galerkin method in (
37). We chose the wave number
(or
) and the exact solution
and
where
denotes the outer normal vector at a point on the surface
. We can write each term of Equation (
26) as:
and
Then, from (
26), (
54) and (
55), the following holds.
We used different values of
for our investigation. In
Table 1, we present the results of the errors in energy norm and
-norm for
for the uniform
h version with polynomial degree
. In
Figure 1 and
Figure 2, we compare the
h-version with different
. The exact norm, known by extrapolation, for
is
, for
is
, and for
is
. Here,
and
(see [
27]). The exact
-norms, known by extrapolation, for
are
and
; for
are
and
; and for
are
and
.
The convergence rates , for are, for the energy norm , and for the -norm and . With , the energy norm of , the -norms of and and , for the energy norm , and for -norm and .
Let us compare the numerical convergence rates above for the boundary element methods obtained in the above example with the theoretical convergence rates predicted by Theorem 1. Note that we have implemented the boundary integral equation system (
26), and (
27) and note the strongly elliptic system (
30), where convergence is guaranteed due to Theorem 1. Nevertheless, our experiments show convergence for the boundary element solution, but with suboptimal convergence rates. Theorem 1 predicts (when Raviart–Thomas elements are used to approximate
and piecewise linear elements to approximate
M) a convergence rate of order
in the energy norm for smooth solutions
and
M. Our computations depend on the parameter
which is a well-known effect with boundary integral equations where it may come to spurious eigenvalues diminishing the orders of the Galerkin approximations. Due to the cube
, the numerical solution might become singular near the edges and corners of
; hence, the Galerkin scheme converges sub-optimally.
Next, we applied the boundary element method above to compute the first terms in the asymptotic expansion of the electrical field considered in
Section 1 (Remark 1). In this way we obtained good results for the electrical field at some point away from the transmission surface
by only computing a few terms in the expansion.
Algorithm for the asymptotic of the eddy current problem:
We have
, and calculate the error
,
, where
,
and
. To find
, Equations (
25)–(
53) are used. We present the results in
Table 2 and in
Figure 3.